Total variation approximation for quasi-equilibrium distributions

TL;DR

This paper introduces a method to approximate quasi-stationary distributions in biological models using total variation, ensuring existence, uniqueness, and computational simplicity under certain conditions.

Contribution

It provides biologically plausible conditions for the existence and uniqueness of quasi-stationary distributions and offers a practical approximation method.

Findings

Conditions for uniqueness of quasi-stationary distributions

Total variation approximation method developed

Simplifies computation of long-term population behavior

Abstract

Quasi-stationary distributions, as discussed by Darroch & Seneta (1965), have been used in biology to describe the steady state behaviour of population models which, while eventually certain to become extinct, nevertheless maintain an apparent stochastic equilibrium for long periods. These distributions have some drawbacks: they need not exist, nor be unique, and their calculation can present problems. In this paper, we give biologically plausible conditions under which the quasi-stationary distribution is unique, and can be closely approximated by distributions that are simple to compute.